nprmdvdsfacm1lem4 - Mathbox for Alexander van der Vekens

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Theorem nprmdvdsfacm1lem4 48528

Description: Lemma 4 for nprmdvdsfacm1 48529. (Contributed by AV, 7-Apr-2026.)

**Assertion**
Ref	Expression
nprmdvdsfacm1lem4	⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝑁 ∥ (!‘(𝑁 − 1)))

**Proof of Theorem nprmdvdsfacm1lem4**
Step	Hyp	Ref	Expression
1		eluzelz 12900	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘6) → 𝑁 ∈ ℤ)
2	1	3ad2ant1 1151	. 2 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝑁 ∈ ℤ)
3		elfzoelz 13716	. . . 4 ⊢ (𝐴 ∈ (2..^𝑁) → 𝐴 ∈ ℤ)
4		id 23	. . . . 5 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℤ)
5		2z 12653	. . . . . . 7 ⊢ 2 ∈ ℤ
6	5	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℤ → 2 ∈ ℤ)
7	6, 4	zmulcld 12734	. . . . 5 ⊢ (𝐴 ∈ ℤ → (2 · 𝐴) ∈ ℤ)
8	4, 7	zmulcld 12734	. . . 4 ⊢ (𝐴 ∈ ℤ → (𝐴 · (2 · 𝐴)) ∈ ℤ)
9	3, 8	syl 18	. . 3 ⊢ (𝐴 ∈ (2..^𝑁) → (𝐴 · (2 · 𝐴)) ∈ ℤ)
10	9	3ad2ant2 1152	. 2 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (𝐴 · (2 · 𝐴)) ∈ ℤ)
11		1z 12651	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
12		6nn 12357	. . . . . . . . . . 11 ⊢ 6 ∈ ℕ
13	12	nnzi 12645	. . . . . . . . . 10 ⊢ 6 ∈ ℤ
14		1re 11235	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
15		6re 12358	. . . . . . . . . . 11 ⊢ 6 ∈ ℝ
16		1lt6 12455	. . . . . . . . . . 11 ⊢ 1 < 6
17	14, 15, 16	ltleii 11360	. . . . . . . . . 10 ⊢ 1 ≤ 6
18		eluz2 12896	. . . . . . . . . 10 ⊢ (6 ∈ (ℤ_≥‘1) ↔ (1 ∈ ℤ ∧ 6 ∈ ℤ ∧ 1 ≤ 6))
19	11, 13, 17, 18	mpbir3an 1360	. . . . . . . . 9 ⊢ 6 ∈ (ℤ_≥‘1)
20		uzss 12913	. . . . . . . . 9 ⊢ (6 ∈ (ℤ_≥‘1) → (ℤ_≥‘6) ⊆ (ℤ_≥‘1))
21	19, 20	ax-mp 5	. . . . . . . 8 ⊢ (ℤ_≥‘6) ⊆ (ℤ_≥‘1)
22	21	sseli 3930	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘6) → 𝑁 ∈ (ℤ_≥‘1))
23		elnnuz 12930	. . . . . . 7 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ_≥‘1))
24	22, 23	sylibr 237	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘6) → 𝑁 ∈ ℕ)
25	24	3ad2ant1 1151	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝑁 ∈ ℕ)
26		nnm1nn0 12572	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
27	25, 26	syl 18	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (𝑁 − 1) ∈ ℕ₀)
28	27	faccld 14350	. . 3 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (!‘(𝑁 − 1)) ∈ ℕ)
29	28	nnzd 12644	. 2 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (!‘(𝑁 − 1)) ∈ ℤ)
30		nprmdvdsfacm1lem1 48525	. 2 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝑁 ∥ (𝐴 · (2 · 𝐴)))
31		elfzo2 13719	. . . . . 6 ⊢ (𝐴 ∈ (2..^𝑁) ↔ (𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐴 < 𝑁))
32		2eluzge1 12934	. . . . . . . . . 10 ⊢ 2 ∈ (ℤ_≥‘1)
33		uzss 12913	. . . . . . . . . 10 ⊢ (2 ∈ (ℤ_≥‘1) → (ℤ_≥‘2) ⊆ (ℤ_≥‘1))
34	32, 33	ax-mp 5	. . . . . . . . 9 ⊢ (ℤ_≥‘2) ⊆ (ℤ_≥‘1)
35	34	sseli 3930	. . . . . . . 8 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ (ℤ_≥‘1))
36	35	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐴 < 𝑁) → 𝐴 ∈ (ℤ_≥‘1))
37	5	a1i 11	. . . . . . . . 9 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 2 ∈ ℤ)
38		eluzelz 12900	. . . . . . . . 9 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ ℤ)
39	37, 38	zmulcld 12734	. . . . . . . 8 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (2 · 𝐴) ∈ ℤ)
40	39	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐴 < 𝑁) → (2 · 𝐴) ∈ ℤ)
41		1lt2 12440	. . . . . . . 8 ⊢ 1 < 2
42		eluz2 12896	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 2 ≤ 𝐴))
43		zre 12622	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
44	43	3ad2ant2 1152	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 2 ≤ 𝐴) → 𝐴 ∈ ℝ)
45		2re 12342	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
46	45	a1i 11	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 2 ≤ 𝐴) → 2 ∈ ℝ)
47		2pos 12372	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
48		0red 11238	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℤ → 0 ∈ ℝ)
49	45	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℤ → 2 ∈ ℝ)
50	48, 49, 43	3jca 1146	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℤ → (0 ∈ ℝ ∧ 2 ∈ ℝ ∧ 𝐴 ∈ ℝ))
51		ltletr 11329	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 2 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((0 < 2 ∧ 2 ≤ 𝐴) → 0 < 𝐴))
52	50, 51	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℤ → ((0 < 2 ∧ 2 ≤ 𝐴) → 0 < 𝐴))
53	47, 52	mpani 709	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℤ → (2 ≤ 𝐴 → 0 < 𝐴))
54	53	imp 412	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 2 ≤ 𝐴) → 0 < 𝐴)
55	54	3adant1 1148	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 2 ≤ 𝐴) → 0 < 𝐴)
56	44, 46, 55	3jca 1146	. . . . . . . . . . 11 ⊢ ((2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 2 ≤ 𝐴) → (𝐴 ∈ ℝ ∧ 2 ∈ ℝ ∧ 0 < 𝐴))
57	42, 56	sylbi 220	. . . . . . . . . 10 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (𝐴 ∈ ℝ ∧ 2 ∈ ℝ ∧ 0 < 𝐴))
58	57	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐴 < 𝑁) → (𝐴 ∈ ℝ ∧ 2 ∈ ℝ ∧ 0 < 𝐴))
59		ltmulgt12 12102	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 2 ∈ ℝ ∧ 0 < 𝐴) → (1 < 2 ↔ 𝐴 < (2 · 𝐴)))
60	58, 59	syl 18	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐴 < 𝑁) → (1 < 2 ↔ 𝐴 < (2 · 𝐴)))
61	41, 60	mpbii 236	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐴 < 𝑁) → 𝐴 < (2 · 𝐴))
62	36, 40, 61	3jca 1146	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐴 < 𝑁) → (𝐴 ∈ (ℤ_≥‘1) ∧ (2 · 𝐴) ∈ ℤ ∧ 𝐴 < (2 · 𝐴)))
63	31, 62	sylbi 220	. . . . 5 ⊢ (𝐴 ∈ (2..^𝑁) → (𝐴 ∈ (ℤ_≥‘1) ∧ (2 · 𝐴) ∈ ℤ ∧ 𝐴 < (2 · 𝐴)))
64		elfzo2 13719	. . . . 5 ⊢ (𝐴 ∈ (1..^(2 · 𝐴)) ↔ (𝐴 ∈ (ℤ_≥‘1) ∧ (2 · 𝐴) ∈ ℤ ∧ 𝐴 < (2 · 𝐴)))
65	63, 64	sylibr 237	. . . 4 ⊢ (𝐴 ∈ (2..^𝑁) → 𝐴 ∈ (1..^(2 · 𝐴)))
66	65	3ad2ant2 1152	. . 3 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝐴 ∈ (1..^(2 · 𝐴)))
67	11	a1i 11	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 1 ∈ ℤ)
68	5	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ (2..^𝑁) → 2 ∈ ℤ)
69	68, 3	zmulcld 12734	. . . . . . 7 ⊢ (𝐴 ∈ (2..^𝑁) → (2 · 𝐴) ∈ ℤ)
70	69	3ad2ant2 1152	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (2 · 𝐴) ∈ ℤ)
71	47	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℤ → 0 < 2)
72		lemul2 12095	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (2 ≤ 𝐴 ↔ (2 · 2) ≤ (2 · 𝐴)))
73	49, 43, 49, 71, 72	syl112anc 1401	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → (2 ≤ 𝐴 ↔ (2 · 2) ≤ (2 · 𝐴)))
74		1red 11236	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (2 · 2) ≤ (2 · 𝐴)) → 1 ∈ ℝ)
75		2t2e4 12431	. . . . . . . . . . . . . . 15 ⊢ (2 · 2) = 4
76		4re 12352	. . . . . . . . . . . . . . 15 ⊢ 4 ∈ ℝ
77	75, 76	eqeltri 2858	. . . . . . . . . . . . . 14 ⊢ (2 · 2) ∈ ℝ
78	77	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (2 · 2) ≤ (2 · 𝐴)) → (2 · 2) ∈ ℝ)
79	7	zred 12728	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℤ → (2 · 𝐴) ∈ ℝ)
80	79	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (2 · 2) ≤ (2 · 𝐴)) → (2 · 𝐴) ∈ ℝ)
81		1lt4 12446	. . . . . . . . . . . . . . . 16 ⊢ 1 < 4
82	14, 76, 81	ltleii 11360	. . . . . . . . . . . . . . 15 ⊢ 1 ≤ 4
83	82, 75	breqtrri 5136	. . . . . . . . . . . . . 14 ⊢ 1 ≤ (2 · 2)
84	83	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (2 · 2) ≤ (2 · 𝐴)) → 1 ≤ (2 · 2))
85		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (2 · 2) ≤ (2 · 𝐴)) → (2 · 2) ≤ (2 · 𝐴))
86	74, 78, 80, 84, 85	letrd 11394	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ (2 · 2) ≤ (2 · 𝐴)) → 1 ≤ (2 · 𝐴))
87	73, 86	sylbida 604	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 2 ≤ 𝐴) → 1 ≤ (2 · 𝐴))
88	87	3adant1 1148	. . . . . . . . . 10 ⊢ ((2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 2 ≤ 𝐴) → 1 ≤ (2 · 𝐴))
89	42, 88	sylbi 220	. . . . . . . . 9 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 1 ≤ (2 · 𝐴))
90	89	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐴 < 𝑁) → 1 ≤ (2 · 𝐴))
91	31, 90	sylbi 220	. . . . . . 7 ⊢ (𝐴 ∈ (2..^𝑁) → 1 ≤ (2 · 𝐴))
92	91	3ad2ant2 1152	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 1 ≤ (2 · 𝐴))
93		eluz2 12896	. . . . . 6 ⊢ ((2 · 𝐴) ∈ (ℤ_≥‘1) ↔ (1 ∈ ℤ ∧ (2 · 𝐴) ∈ ℤ ∧ 1 ≤ (2 · 𝐴)))
94	67, 70, 92, 93	syl3anbrc 1362	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (2 · 𝐴) ∈ (ℤ_≥‘1))
95		zsqcl 14195	. . . . . . 7 ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ)
96	3, 95	syl 18	. . . . . 6 ⊢ (𝐴 ∈ (2..^𝑁) → (𝐴↑2) ∈ ℤ)
97	96	3ad2ant2 1152	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (𝐴↑2) ∈ ℤ)
98		3z 12654	. . . . . . . 8 ⊢ 3 ∈ ℤ
99	98	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 3 ∈ ℤ)
100	3	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝐴 ∈ ℤ)
101		nprmdvdsfacm1lem2 48526	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 3 ≤ 𝐴)
102		eluz2 12896	. . . . . . 7 ⊢ (𝐴 ∈ (ℤ_≥‘3) ↔ (3 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 3 ≤ 𝐴))
103	99, 100, 101, 102	syl3anbrc 1362	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝐴 ∈ (ℤ_≥‘3))
104		2timesltsq 48268	. . . . . 6 ⊢ (𝐴 ∈ (ℤ_≥‘3) → (2 · 𝐴) < (𝐴↑2))
105	103, 104	syl 18	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (2 · 𝐴) < (𝐴↑2))
106	94, 97, 105	elfzod 13720	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (2 · 𝐴) ∈ (1..^(𝐴↑2)))
107		oveq2 7424	. . . . . 6 ⊢ (𝑁 = (𝐴↑2) → (1..^𝑁) = (1..^(𝐴↑2)))
108	107	eleq2d 2848	. . . . 5 ⊢ (𝑁 = (𝐴↑2) → ((2 · 𝐴) ∈ (1..^𝑁) ↔ (2 · 𝐴) ∈ (1..^(𝐴↑2))))
109	108	3ad2ant3 1153	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → ((2 · 𝐴) ∈ (1..^𝑁) ↔ (2 · 𝐴) ∈ (1..^(𝐴↑2))))
110	106, 109	mpbird 260	. . 3 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (2 · 𝐴) ∈ (1..^𝑁))
111		muldvdsfacm1 48277	. . 3 ⊢ ((𝐴 ∈ (1..^(2 · 𝐴)) ∧ (2 · 𝐴) ∈ (1..^𝑁)) → (𝐴 · (2 · 𝐴)) ∥ (!‘(𝑁 − 1)))
112	66, 110, 111	syl2anc 596	. 2 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (𝐴 · (2 · 𝐴)) ∥ (!‘(𝑁 − 1)))
113	2, 10, 29, 30, 112	dvdstrd 16389	1 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝑁 ∥ (!‘(𝑁 − 1)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ⊆ wss 3902 class class class wbr 5107 ‘cfv 6537 (class class class)co 7416 ℝcr 11126 0cc0 11127 1c1 11128 · cmul 11132 < clt 11270 ≤ cle 11271 − cmin 11468 ℕcn 12260 2c2 12322 3c3 12323 4c4 12324 6c6 12326 ℕ₀cn0 12531 ℤcz 12618 ℤ_≥cuz 12890 ..^cfzo 13711 ↑cexp 14127 !cfa 14339 ∥ cdvds 16346

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-n0 12532 df-z 12619 df-uz 12891 df-rp 13045 df-fz 13564 df-fzo 13712 df-seq 14068 df-exp 14128 df-fac 14340 df-bc 14369 df-dvds 16347

This theorem is used by: nprmdvdsfacm1 48529

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